Splitting property in infinite posets (Q1356552)

Given a poset \(P\), a subset \(H\subset P\) is a generator if \(D(H)\cup U(H)=P\), where \(D(H)\) is the ideal (downset) generated by \(H\) and \(U(H)\) is the filter (upset) generated by \(H\). Thus if \(S\) is an antichain which is a generator it is maximal and conversely a maximal antichain is a generator. \(H\) has the splitting property if \(H\) can be partitioned \(H=(H_1,H_2)\) such that \(U(H_1)\cup D(H_2)=P\). Thus, if \(P\) is finite, the collection of all minimal elements yields \(U(H)\cup D(\varnothing)=P\), while the collection of all maximal elements yields \(U(\varnothing)\cup D(H)=P\). If \(P\) is infinite there may be complete failure in the observations above and thus it becomes an interesting and more subtle problem to deal with even the existence of generators \(H\) with special splitting properties in general and more special cases as the author has done, basing his further observation on work by R. Ahlswede and L. H. Khachatrian among others.